antidérivée

An antidérivée, also known as an antiderivative or primitive function, is a function whose derivative is a given function. In simpler terms, if you have a function f(x), an antidérivée of f(x) is a function F(x) such that the derivative of F(x) with respect to x equals f(x).

The process of finding an antidérivée is called antidifferentiation or integration. For any given function f(x), there are infinitely many antidérivées, all differing by a constant. This constant is often denoted by 'C' and is called the constant of integration. If F(x) is one antidérivée of f(x), then F(x) + C is also an antidérivée for any real number C.

The concept of antidérivées is fundamental to calculus and is directly related to the process of integration. The Fundamental Theorem of Calculus establishes a crucial link between differentiation and integration, stating that integration and differentiation are inverse operations. This means that if you integrate a function and then differentiate the result, you get back the original function (up to a constant).

For example, if f(x) = 2x, then an antidérivée is F(x) = x^2, because the derivative of x^2 is 2x. Other antidérivées include x^2 + 1, x^2 - 5, and so on. The general form of the antidérivée for f(x) = 2x is x^2 + C.

Antidérivées are used in various applications, including calculating areas under curves, solving differential equations, and determining displacement from velocity.